THE inverse tangent function is an elementary mathematical

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1 A Sharp Double Inequality for the Inverse Tangent Function Gholamreza Alirezaei arxiv: v [cs.it] 8 Jul 03 Abstract The inverse tangent function can be bounded by different inequalities, for eample by Shafer s inequality. In this publication, we propose a new sharp double inequality, consisting of a lower an upper bound, for the inverse tangent function. In particular, we sharpen Shafer s inequality calculate the best corresponding constants. The maimum relative errors of the obtained bounds are approimately smaller than 0.7% 0.3% for the lower upper bound, respectively. Furthermore, we determine an upper bound on the relative errors of the proposed bounds in order to describe their tightness analytically. Moreover, some important properties of the obtained bounds are discussed in order to describe their behavior achieved accuracy. Inde Terms trigonometric bounds; Shafer s inequality; inverse tangent approimation; I. INTRODUCTION THE inverse tangent function is an elementary mathematical function that appears in many applications, especially in different fields of engineering. In electrical engineering, especially in the communication theory signal processing, it is mostly used to describe the phase of a comple-valued signal. But there are many other applications in which the inverse tangent function plays an important role. On the one h, it is often used as an approimation for more comple functions because of its elementary behavior. For instance, the Heaviside step function is the most famous function that can be very accurately approimated by the inverse tangent function. On the other h, it is sometimes approimated by simpler functions in order to enable further calculations. For instance, the inverse tangent function can be accurately approimated by its argument if the absolute value of the argument is sufficiently small. Quite naturally the problem arises how to replace the inverse tangent function with a surrogate function, in order to approimate the inverse tangent function as well as other contemplable functions accurately. If a surrogate function with a mathematically simple form could be found, then the subsequent application of such a surrogate function would be considerable. A few application cases are in the field of information estimation theory where an unknown phase shift or the direction of arrival is estimated, for eample by using the CORDIC-algorithm [], the MUSIC-algorithm [], or MAP ML estimators [3]. Some other cases are in the field of system design control theory where a non-linear network unit is modeled by a non-linear function, for instance G. Alirezaei is with the Institute for Theoretical Information Technology, RWTH Aachen University, 5056 Aachen, Germany ( alirezaei@ti.rwthaachen.de). The present work is categorized in terms of Mathematics Subject Classification (MSC00): 6D05, 6D07, 6D5, 33B0, 39B6. the saturation behavior of an amplifier [] or the sigmoidal non-linearity in neuronal networks [5]. Some more application cases are related to the theory of signals systems, where a signal should be mapped into a set of coefficients of basis functions, however the transformation is not feasible because of the phase description by the inverse tangent function, for eample in some Fourier-related transforms [6]. Many other applications are likewise conceivable. But we have to mention that finding a simple replacement for the inverse tangent function is in fact difficult. In the present work, we thus focus only on a special idea which has some nice properties is described in the following. In [7], R. E. Shafer proposed the elementary problem: Show that for all > 0 the inequality 3 < arctan() () holds, where arctan() denotes the inverse tangent function that is defined for all real numbers. From Shafar s problem several inequalities have been emerged to date. In particular, the authors in [8] investigated double inequalities of the form a a < arctan() < b b, > 0, () they determined the coefficients a, a, b b such that the above double inequality is sharp. In the present work, we follow a similar idea investigate a generalized version of the double inequality in (). We consider functions of the type c c c 3 (3) with positive real coefficients c, c c 3, because such kind of functions has advantageous properties in order to replace the inverse tangent function as we will discuss in the net section. Then we determine the triple (c, c, c 3 ) such that a lower an upper bound for the inverse tangent function is achieved. In order to describe the tightness of the obtained bounds, we determine an upper bound on the relative errors of the proposed bounds. Furthermore, we discuss some corresponding properties of the proposed bounds visualize the achieved results. G. A. July 8, 03 Mathematical Notations: Throughout this paper we denote the set of real numbers by R. The mathematical operation denotes the absolute value of any real number. Furthermore, O ( ω() ) denotes the order of any function ω().

2 II. MAIN THEOREMS In the current section, we present the new bounds for the inverse tangent function describe some of their important properties. Theorem II. For all R, let, h() be defined by := ( ), () := arctan() (5) h() := ( 6 6 ). (6) Then, for all R, the double inequality holds. h() (7) Proof: See Appendi A. Remark II. The functions, h() are point symmetric such that the identities f( ) =, g( ) = h( ) = h() hold. Hence, it is sufficient to consider only the case of 0. ( Remark II.3 The triples ) ( ), ( ), 6, ( 6 ), are the best possible ones such that the above double inequality holds. In other words, no component of the first triple can be replaced by a smaller value no component of the second triple can be replaced by a larger value with respect to 0 while keeping the other components fied. In this sense, the double inequality in Theorem II. is sharp. We get a first impression of the nature of the bounds from Figure. As we can see the double inequality in Theorem II. is very tight. The curves seem to be continuous, strictly increasing conve. Hence, we elaborately discuss the mathematical properties of the obtained bounds in the following. On the one h, the first three elements in the Taylor series epansions of, h() as approaches zero are obtained as ( ) 3 5 O ( 7), (8) O ( 7) (9) h() O ( 7). (0) Remark II. Only the both first elements in the Taylor series epansions of are identical to each other, while in the Taylor series epansions of h() the both h() Fig.. The inverse tangent function its bounds from Theorem II. are visualized for the range of 0 0. The curves are closely adjacent to one another such that without magnification the differences are not really visible. The curves are equal at zero approach the same upper limit as approaches infinity. first two elements are pairwise identical to each other. Thus, h() achieves a better approimation of than for sufficiently small. On the other h, the first three elements in the asymptotic power series epansions of, h() as approaches infinity are obtained as O ( 3), () 3 3 O( 5) () h() O ( 3) (3) by using the general definition of the asymptotic power series epansion [9, p., Definition.3.3] simple calculations. Remark II.5 The both first two elements in the asymptotic power series epansions of are pairwise identical to each other while in the asymptotic power series epansions of h() only the both first elements are identical to each other. Thus, achieves a better approimation of than h() for sufficiently large. Corollary II.6 From equations (8) (3) we conclude that lim = lim = lim h() = 0 () ±0 ±0 ±0 lim = ± lim = ± lim h() = ± ±. (5) Lemma II.7 For all R, both bounds h() are continuous.

3 3 Proof: Both numerators denominators of h() are continuous functions in the denominators are always non-zero which imply the absence of discontinuities. Lemma II.8 For all R, both bounds h() are strictly increasing. Proof: By differentiation we obtain the following first derivative d d c c c 3 = c c c c 3 c c 3 [ c c c 3 ]. (6) This derivative is positive for all R because c, c c 3 are positive constants in both bounds. Hence, the bounds are strictly increasing. Corollary II.9 Both bounds h() are differentiable on R, because the first derivative of the bounds eists due to the derivative in equation (6). Corollary II.0 Both bounds h() are limited, due to equation (5) because of the monotonicity in Lemma II.8. Corollary II. Both bounds h() do not have any critical points, because they are strictly increasing differentiable on R. Lemma II. For all 0, both bounds h() are concave while for all 0, both bounds are conve. Proof: By differentiation we obtain the following second derivative d d c c c 3 = c 3 3c c c c 3 3c c c 3 [ (c c 3 ) 3 / c c c 3 ] 3. (7) The sign of this derivative is only dependent on because c, c c 3 are positive constants in both bounds. Hence, this derivative is non-positive for all 0 non-negative for all 0 which completes the proof. Corollary II.3 Both bounds h() have the same unique inflection point at the origin, due to opposing conveities for 0 0, see Lemma II.. In the following enumeration, we now summarize the properties of the bounds that have been shown, thus far. ) The bounds are equal only at zero they approach the same limit as approaches infinity. ) Both bounds are point symmetric, continuous, strictly increasing, differentiable, limited. 3) They are conve for all 0 concave otherwise. ) There are no critical points. 5) Both bounds have the same unique inflection point h() f () f () r f () r h () h() f () h()f () Fig.. The relative errors of the bounds for the inverse tangent function are visualized for the range of 0 0. All curves are equal at zero they approach zero as approaches infinity. r h () is smaller, hence, better than r f () for sufficiently small values of, while for sufficiently large values of the opposite holds. The relative error r f () is approimately smaller than 0.7% while r h () is approimately smaller than 0.3%. The ratio [h() ]/[h()] lies between r f () r h (), hence, is an upper bound for min{r f (), r h ()} while []/ is an upper bound for ma{r f (), r h ()}. The above properties enable us to use the proposed bounds suitable in future works. It remains to show the tightness of the bounds with respect to the inverse tangent function. For this purpose we deduce an upper-bound on the actual relative errors of the bounds, in the following. Definition II. For all R, the relative errors of the bounds given in Theorem II. are defined by r f () := r h () := h() (8). (9) Note that = 0 is a removable singularity for both last ratios because of approimations (8), (9) (0). Thus, r f () r h () are continuously etendable over = 0. All following fractions are also continuously etendable over = 0 in a similar manner so that no difficulties related to singularities occur, hereinafter. Theorem II.5 For all R, the inequalities ma { r f (), r h () } = 0 9 ( ) 6 (0) 9

4 min { r f (), r h () } = 0 9 ( ) 9 ( ) hold. ma { r f (), r h () } () Proof: See Appendi B. Note, that the inequalities in Theorem II.5 do not contain the inverse tangent function, at all. In Figure, the relative errors of the obtained bounds are shown. The maimum relative errors of the bounds are approimately smaller than 0.7% 0.3% for h(), respectively. It is worthwhile mentioning that both bounds are valid for the whole domain of real numbers. III. CONCLUSION In the present work, we have investigated the approimation of the inverse tangent function deduced two new bounds. We have derived a lower an upper bound with simple closed-form formulae which are sharp very accurate. Furthermore, we have presented some useful important properties of the obtained bounds. These properties can be necessary in future works. Moreover, we have investigated the relative errors of the proposed bounds. The corresponding maimum relative errors of the bounds are approimately smaller than 0.7% 0.3% for the lower bound upper bound, respectively. These values show that the obtained bounds are very accurate thus are suitably applicable in the most engineering problems. Finally, we have illustrated some results in order to visualize the achieved gains. APPENDIX A PROOF OF THE BOUNDS Lemma A. The transcendental number can be bounded by the double inequality 9 3 < < 0. () Proof: Both bounds are well known for long, see for eample [0]. A new proof of the upper bound can be found in []. We here give an elementary proof of the lower bound. The identities = k= k(k) ζ() = k= k = 6, see for eample [, p. 8, eq p., eq. 0..3], are used to deduce 6 = k= k k= k(k ) = k= k (k ) = }{{ 36 } k (k ) > 9 8. (3) k= = 9 8 Hence 9 3 < follows. In the following, we denote the differences h() by f () := () h () := h(), (5) respectively. From Corollary II.6 it is immediately deduced that lim f () = lim h() ±0 ±0 = lim f () = lim h() = 0. (6) ± ± By direct algebra the first derivatives of f () h () are given as d f () d d h () d = ( ) ( ) ( ( ) ) (7) = ( 6 6 ) ( 6 ) ( 6 respectively. ( 6 ) ), (8) Corollary A. The first derivatives of f () h () vanish only at three real points, namely { f 0, ± ( ) } (9) 6 respectively. h { } , ± 576 (0, (30) ) Proof: We set (7) (8) equal to zero obtain the points in (9) (30) by direct calculations. It remains to prove that all points are real. This is done by showing that the discriminant functions y f (ν) := ν 36ν 60 = (ν 8)(0 ν ) (3) y h (ν) := 5ν 08ν 576 = (5ν 8)( ν ) (3) are non-negative for ν =. A curve tracing of y f (ν) y h (ν) leads to the relationships y f (ν) 0 8 ν 0 (33) y h (ν) ν, (3)

5 5 respectively. Hence, both y f (ν) y h (ν) are non-negative for all 8 5 ν 0. By comparing the latter double inequality with the double inequality in Lemma A. we deduce that y f () y h () are non-negative, hence, all roots in (9) (30) are real. Corollary A.3 The difference f () is positive for all sufficiently small positive real numbers. For all negative real numbers with sufficiently small absolute value, the difference f () is negative. Proof: We incorporate the equations (8) (9) into () to derive the first-order approimation of f () as f () 0 3( ) 3 O ( 5) (35) for all sufficiently small values of. From the double inequality in Lemma A., we deduce that the last ratio is always positive which completes the proof. Corollary A. The difference h () is positive for all sufficiently large positive real numbers. For all negative real numbers with sufficiently large absolute value, the difference h () is negative. Proof: We incorporate the equations () (3) into (5) to derive the first-order asymptotic approimation of h () as h () 0 O ( ) (36) for all sufficiently large values of. From the double inequality in Lemma A., we deduce that the last ratio is always positive which completes the proof. Proof of THEOREM II.: We only consider the case of 0. The case of 0 can be proved analogously, due to the point symmetric property of all functions in Theorem II.. On the one h, we know from Corollary A. that each of differences f () h () has only one stationary point for all > 0. On the other h, each of them attains equal values at = 0 as, i.e., f (0) = f ( ) = 0 h (0) = h ( ) = 0, according to the equation (6). Hence, because of Corollary A.3 A., as increases from zero to infinity each of the differences f () h () increases monotonically from zero to a maimum value from there on decreases monotonically toward zero. Thus, both differences f () h () are non-negative for all 0. In other words, if one of the differences had at least one sign change for some value of > 0, then it would have at least two stationary points for > 0, but this contradicts the curve tracing in Corollary A.. APPENDIX B PROOF OF THE RELATIVE ERRORS Definition B. Let r f () r h () be defined as in Definition II.. Then, we define two auiliary sets by I f := { R r f () r h () } (37) I h := { R r f () < r h () }. (38) Note that both sets I f I h are disjoint their union is the whole real domain. Corollary B. For all I f, 0, the inequality holds. If I h, 0, then the inequality (39) < (0) holds. In the case of I f with < 0 I h with < 0 the above inequalities are reversed. Proof: For all I f with 0, from Definition II. B. it follows that r f () r h () r f () r h () h(). () Similarly, for all I h with 0 it follows that r f () < r h () r f () < r h () < h() <. () In the case of < 0, the functions, h() are negative, hence, the inequalities in (39) (0) are reversed. Proof of THEOREM II.5: The proof of inequality (0) follows from inequality (7) Definition II.. It gives r f () = r h () = h() which in turn result in ma { r f (), r h () } (3) (). (5) The proof of inequality () follows from Definition II. Corollary B.. For all I f with 0, it gives r f () = r h () = h() which in turn result in r h () h() h() h() = = (6) (7) r f (). (8)

6 6 If I h with 0, then it gives r f () = < r h () = h() > which in turn result in r f () < h() h() h() = = (9) (50) < r h(). (5) From the double inequalities (8) (5), we deduce that min { r f (), r h () } ma{ r f (), r h () } (5) for all 0. For the case of < 0, the proof can be obtained analogously. The identities in (0) () arise from straightforward calculations. ACKNOWLEDGMENT Research described in the present work was supervised by Univ.-Prof. Dr. rer. nat. R. Mathar, Institute for Theoretical Information Technology, RWTH Aachen University. The author would like to thank him for his professional advice patience. REFERENCES [] J. E. Volder, The cordic trigonometric computing technique, Electronic Computers, IRE Transactions on, vol. EC-8, no. 3, pp , 959. [] R. Schmidt, Multiple emitter location signal parameter estimation, Antennas Propagation, IEEE Transactions on, vol. 3, no. 3, pp , 986. [3] H. Fu P.-Y. Kam, MAP/ML estimation of the frequency phase of a single sinusoid in noise, Signal Processing, IEEE Transactions on, vol. 55, no. 3, pp , 007. [] X.-B. Zeng, Q.-M. Hu, J.-M. He, Q.-P. Tu, X.-J. Yu, High power RF amplifier s new nonlinear models, in Microwave Conference Proceedings, 005. APMC 005. Asia-Pacific Conference Proceedings, vol., 005. [5] B. Widrow M. Lehr, 30 years of adaptive neural networks: perceptron, madaline, backpropagation, Proceedings of the IEEE, vol. 78, no. 9, pp. 5, 990. [6] C. Hwang H.-C. Chow, Simplification of z-transfer function via Padé approimation of tangent phase function, Automatic Control, IEEE Transactions on, vol. 30, no., pp. 0 0, 985. [7] R. E. Shafer, Elementary problems: E 867, The American Mathematical Monthly, vol. 73, no. 3, p. 309, 966. [8] F. Qi, S.-Q. Zhang, B.-N. Guo, Sharpening generalizations of Shafer s inequality for the arc tangent function, Journal of Inequalities Applications, vol. 009, 009. [9] N. Bleistein R. A. Helsman, Asymptotic Epansions of Integrals. New York: Dover Publications, 986. [0] O. Neugebauer, The Eact Sciences in Antiquity. Dover Publications Incorporated, 969. [] N. D. Elkies, Why is so close to 0? The American Mathematical Monthly, vol. 0, no. 7, p. 59, 003. [] I. S. Gradshteyn I. M. Ryzhik, Table of Integrals, Series, Products, 7th ed. London: Academic Press, 007.